Do you really expect a logical proof? If so, I think you’ve misunderstood logic.
Are you looking for a proof using a logical system containing some rules of inference, starting from some axioms, leading to the conclusion that “God exists”? That’s fruitless. The theorems of a system of logic (the things that can be proven from the axioms according to the rules of inference) are contained completely in the axioms and rules. There’s no way for anything “new” to get into the system. Thus, you can only “prove” that there is a God if you first assume there is a God. Most people don’t find that very compelling.
Another difficulty is incompleteness. Since 1931, we’ve given up on being able to “prove” everything that is true in a logical system. Any system powerful enough to express arithmetic is either inconsistent or incomplete. We’re usually not interested in inconsistent systems, which means we’ll have to accept incomplete systems. A logical system is incomplete if it contains truths that cannot be proven. You can’t say that because a thing is unprovable, it is necessarily untrue.
Why would you choose to not believe in God, unless his nonexistence were proven logically? Think carefully before you trot out “you can’t prove a negative”. Euclid nicely proved that in plane geometry, there does not exist a triangle has angles that add up to 181 degrees. Besides, when two propositions are each other’s negation, who gets to say that one is “the proposition” and the other is its negation?
Perhaps you are really talking about scientific as opposed to logical knowledge. In that case, you should consider that there are other ways of knowing besides the scientific method. You might even consider some of them “valid” or “rational”.
For instance, the scientific method tends to approach statements like “God exists” with maximum skepticism. We cannot assume it is true until we have evidence that it is true, which results in a hostility toward such statements. However, there are many real world situations where maximum skepticism denies knowledge to some truths. If you were interested in finding out whether I liked you, and you approached the question from a position of assuming that I did not like you, you might never discover that I do, in fact, like you.
Applying a standard of “logical” or “scientific” knowledge in arenas where it is not suited could itself be called irrational.
kg m²/s²